Affiliation:
1. Institute of Computer Science, University of Wrocław, Poland
Abstract
We solve open problems concerning the Kleene star
of a finite set
of words over an alphabet
. The
Frobenius monoid
problem is the question for a given finite set of words
, whether the language
is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets
such that the length of the longest words not in
(when
is cofinite) is exponential in the length of the longest words in
and subexponential in the sum of the lengths of words in
. The
factor universality
problem is the question for a given finite set of words
, whether every word over
is a factor (substring) of some word from
. We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets
such that the length of the shortest words not being a factor of any word in
is exponential in the length of the longest words in
and subexponential in the sum of the lengths of words in
. This essentially settles in the negative the longstanding Restivo’s conjecture (1981) and its weak variations. All our solutions are based on one shared construction, and as an auxiliary general tool, we introduce the concept of
set rewriting systems
. Finally, we complement the results with upper bounds.
Funder
National Science Centre, Poland
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
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